A theory of measurement with applications to spectrometry

Abstract

An analytic function operating on experimental data is optimized for making the most accurate measurement of a parameter of the expected data, under the conditions of non-stationary shot noise. The value of the parameter is obtained by equating the operation on experimental data with the same operation on the expected data which contains the parameter as an unknown value. If the operation is represented by a weighting function, the form of the optimum weighting function depends on the initial transformation of the experimental data by the measuring instrument. The optimum weighting function always contains the derivative of the expected signal with respect to the unknown parameter, divided by the time-dependent variance of the received signal. Weighting functions for the logarithmic output of a spectrophotometer are described. The superiority over least-squares curve-matching is shown. A method for determination of peak position by optimum slope measurement is derived. In general, the optimum weighting function is not a matched filter. The optimum result is the same for center-of-gravity measurements. The optimum parameter measurement is equivalent to a least-squares error minimization weighted by the inverse variance or mean-square noise level. This variance weighting is significant in photometric measurements limited by shot noise or other measurements described by Poisson statistics, such that the mean-square noise level varies with time. The form of the optimum filter, for non-white, non-stationary noise is derived.